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Strong convergence theorems for nonlinear operator equations with total quasi-ϕ-asymptotically nonexpansive mappings and applications

  • Xiong-rui Wang1,
  • Shih-sen Chang2Email author,
  • Lin Wang2,
  • Yong-Kun Tang2 and
  • Yu Guang Xu3
Fixed Point Theory and Applications20122012:34

https://doi.org/10.1186/1687-1812-2012-34

Received: 25 September 2011

Accepted: 1 March 2012

Published: 1 March 2012

Abstract

The purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases, and then by using the hybrid algorithm to introduce a new iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, the set of common fixed points of a countable family of total quasi-ϕ-asymptotically nonexpansive mappings and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. As an application, we shall utilize our results to study the iterative solutions of the nonlinear Hammerstian type equation. The results presented in the article improve and extend the corresponding results announced by some authors.

AMS (MOS) subject classification: 47H06; 47H09; 47J05; 47J25.

Keywords

total quasi-ϕ-asymptotically nonexpansive mappingsgeneralized mixed equilibrium problemsquasi-ϕ-asymptotically nonexpansive mappingsrelatively nonexpansive mappingvariational inequalitiesHammerstian type equationfixed point

1 Introduction

Throughout this article, we assume that E is a real Banach space with a dual E*, C is a nonempty closed convex subset of E and 〈·, ·〉 is the duality pairing between members of E and E*, R is the set of all real numbers. In the sequel, we denote by x n x and x n x the weak convergence and strong convergence of sequence {x n }, respectively. The mapping J : E 2 E * defined by
J ( x ) = f * E * : x , f * = x 2 ; f * = x , x E ,

is called the normalized duality mapping.

Let F : C × C R be a bifunction, B: CE* be a nonlinear mapping and Ψ : C ¯ be a proper extended real-valued function. The "so called" generalized mixed equilibrium problem (MEP) for F, B, Ψ is to find x* C such that
F ( x * , y ) + y - x * , B x * + Ψ ( y ) - Ψ ( x * ) 0 , y C .
(1.1)
The set of solutions of (1.1) is denoted by GMEP(F, B, Ψ), i.e.,
GMEP ( F , B , Ψ ) = { x * C : F ( x * , y ) + y x * , B x * + Ψ ( y ) Ψ ( x * ) 0 , y C } .

1.1 Special examples

  1. (1)
    If B = 0, Ψ = 0, then problem (1.1) is reduced to the equilibrium problem (EP), and the set of its solutions is denoted by
    EP ( F ) = { x * C : F ( x * , y ) 0 , y C } .
     
  2. (2)
    If Ψ ≡ 0, then the problem (1.1) is reduced to the generalized equilibrium problem (GEP), and the set of its solutions is denoted by
    GEP ( F , B ) = { x * C : F ( x * , y ) + y - x * , B x * 0 , y C } .
     
  3. (3)
    If B ≡ 0, then the problem (1.1) is reduced to the MEP, and the set of its solutions is denoted by
    MEP ( F , Ψ ) = { x * C : F ( x * , y ) + Ψ ( y ) - Ψ ( x * ) 0 , y C } .
     

These show that the problem (1.1) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.1). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach space (see, for examples [15]).

Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Throughout this article we assume that ϕ : E × E R + is the Lyapunov function which is defined by
ϕ ( x , y ) = x 2 - 2 x , J y + y 2 , x , y E .
Following Alber [6], the generalized projection C : EC is defined by
Π C ( x ) = arg mi n y C ϕ ( y , x ) , x E .

Let S: CC be a mapping and F(S) be the set of fixed points of S.

Recall that a point p C is called an asymptotic fixed point of S, if there exists a sequence {x n } C such that x n p and ||x n - Sx n || → 0. We denoted the set of all asymptotic fixed points of S by F ̃ ( S ) . A point p C is called a strong asymptotic fixed point of S, if there exists a sequence {x n } C such that x n p and ||x n - Sx n || → 0. We denoted the set of all strong asymptotic fixed points of S by F ^ ( S ) .

A mapping S: CC is said to be nonexpansive, if
S x - S y x - y , x , y C .
A mapping S: CC is said to be relatively nonexpansive [7] if, F ( S ) , F ( S ) = F ̃ ( S ) , and
ϕ ( p , S x ) ϕ ( p , x ) , x C , p F ( S ) .
A mapping S: CC is said to be weak relatively nonexpansive [8], if F ( S ) , F ( S ) = F ̃ ( S ) , and
ϕ ( p , S x ) ϕ ( p , x ) , x C , p F ( S ) .

A mapping S: CC is said to be closed, if for any sequence {x n } C with x n x and Sx n y, then Sx = y.

A mapping S: CC is said to be quasi-ϕ-nonexpansive, if F ( S ) and
ϕ ( p , S x ) ϕ ( p , x ) , x C , p F ( S ) .
A mapping S: CC is said to be quasi-ϕ-asymptotically nonexpansive, if F ( S ) and there exists a real sequence {k n } [1, ∞) with k n → 1 such that
ϕ ( p , S n x ) k n ϕ ( p , x ) , n 1 , x C , p F ( S ) .
(1.2)
Definition 1.1 (1) A mapping S: CC is said to be total quasi-ϕ-asymptotically nonexpansive, if F ( S ) and there exist nonnegative real sequences {ν n }, {μ n } with ν n → 0, μ n → 0 (as → ∞) and a strictly increasing continuous function ς : R + R + with ζ(0) = 0 such that for all x C, p F(S)
ϕ ( p , S n x ) ϕ ( p , x ) + ν n ς ( ϕ ( p , x ) ) + μ n , n 1 .
(1.3)
  1. (2)
    A countable family of mappings {S n }: CC is said to be uniformly total quasi-ϕ-asymptotically nonexpansive, if i = 1 F ( S i ) and there exist nonnegative real sequences {ν n }, {μ n } with ν n → 0, μ n → 0 (as n → ∞) and a strictly increasing continuous function ς : R + R + with ζ(0) = 0 such that for for each i > 1, and each x C , p i = 1 F ( S i )
    ϕ ( p , S i n x ) ϕ ( p , x ) + ν n ς ( ϕ ( p , x ) ) + μ n , n 1 .
    (1.4)
     
Remark 1.1 From the definitions, it is easy to know that
  1. (1)

    Each relatively nonexpansive mapping is closed;

     
  2. (2)
    Taking ζ(t) = t, t ≥ 0, ν n = (k n - 1) and μ n = 0, then (1.2) can be rewritten as
    ϕ ( p , S n x ) ϕ ( p , x ) + ν n ς ( ϕ ( p , x ) ) + μ n , n 1 , x C , p F ( S ) .
    (1.5)
     
This implies that each quasi-ϕ-asymptotically nonexpansive mapping must be a total quasi-ϕ-asymptotically nonexpansive mapping, but the converse is not true.
  1. (3)

    The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true;

     
  2. (4)

    The class of quasi-ϕ-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse may be not true;

     
  3. (5)

    The class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.

     
A mapping A: CE* is said to be α-inverse strongly monotone, if there exists α > 0 such that
x - y , A x - A y α A x - A y 2 .

Remark 1.2 If A is an α-inverse strongly monotone mapping, then it is 1 α -Lipschitz continuous.

Iterative approximation of fixed points for relatively nonexpansive mappings in the setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushita and Takahashi [7] obtained some weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Ofoedu and Malonza [4], Zhang [5], Su et al. [8], Zhang and Su [9], Zegeye and Shahzad [10], Wattanawitoon and Kumam [11], Qin et al. [12], Takahashi and Zembayashi [13] extend the notions from relatively nonexpansive mappings, weakly relatively nonexpansive mappings or quasi-ϕ-nonexpansive mappings to quasi-ϕ-asymptotically nonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of quasi-ϕ-nonexpansive mappings or quasi-ϕ-asymptotically nonexpansive mappings.

The purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases, and then by using hybrid algorithm to introduce a new iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a countable family of total quasi-ϕ-asymptotically nonexpansive mappings and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. As an application, we shall utilize our results to study the iterative solutions of the nonlinear Hammerstian type equation. The results presented in the article improve and extend the corresponding results in [15, 716].

2 Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will be needed in proving our main results.

A Banach space E is said to be strictly convex, if x + y 2 < 1 for all x, y U = {z E: ||z|| = 1} with xy. It is said to be uniformly convex, if for each ϵ (0, 2], there exists δ > 0 such that x + y 2 1 - δ for all x, y U with ||x - y|| ≥ ϵ. The convexity modulus of E is the function δ E : (0, 2] → [0, 1] defined by
δ E ( ε ) = inf 1 - 1 2 ( x + y ) : x , y U , x - y ε ,
for all ϵ (0, 2]. It is well known that δ E (ϵ) is a strictly increasing and continuous function with δ E (0) = 0 and δ E ( ε ) ε is nondecreasing for all ϵ (0,2]. Let p > 1, then E is said to be p-uniformly convex, if there exists a constant c > 0 such that δ E (ϵ) ≥ p , ϵ (0, 2]. The space E is said to be smooth, if the limit
lim t 0 x + t y - x t

exists for all x, y U. And E is said to be uniformly smooth, if the limit exists uniformly in x, y U.

In the sequel, we shall make use of the following lemmas.

Lemma 2.1 [17] Let E be a 2-uniformly convex real Banach space, then for all x, y E, the following inequality holds:
x - y 2 c 2 J x - J y ,
(2.1)

where 0 <c ≤ 1, and c is called the 2-uniformly convex constant of E.

Lemma 2.2 [18] Let E be a smooth, strict convex and reflexive Banach space and C be a nonempty closed convex subset of E. Then, the following conclusions hold:
  1. (i)

    ϕ(x, ∏ C y) + ϕ(∏ C y, y) ≤ ϕ (x, y), x C, y E.

     
  2. (ii)
    Let x E and z C, then
    z = Π C x z - y , J x - J z 0 , y C .
     

Lemma 2.3 [18] Let E be a uniformly convex and smooth Banach space and {x n }, {y n } be sequences of E. If ϕ(x n , y n ) → 0 (as n → ∞) and either {x n } or {y n } is bounded, then x n - y n → 0(as n → ∞).

Lemma 2.4 [19] Let E be a uniformly convex Banach space, r be a positive number and B r (θ) be a closed ball of E. For any given points {x1, x2, ... , x n , ...} B r (θ) and for any given positive numbers {λ1, λ2,...} with n = 1 λ n = 1 , then there exists a continuous, strictly increasing and convex function g: [0, 2r) → [0, ∞) with g(0) = 0 such that for any i, j {1, 2,..., }, i <j,
n = 1 λ n x n 2 n = 1 λ n x n 2 - λ i λ j g ( x i - x i ) .
(2.2)

Lemma 2.5 [20] Let E be a smooth, strict convex and reflexive Banach space and C be a nonempty closed convex subset of E , f : C × C R be a bifunction satisfying the following conditions:

(A1) f(x, x) = 0 for all x C;

(A2) f is monotone, i.e., f(x, y) + f(y, x) ≤ 0, x, y C;

(A3) lim supt↓0f(x + t(z - x), y) ≤ f(x, y), x, y, z C;

(A4) The function y f(x, y) is convex and lower semicontinuous.

Then the following conclusions hold:
  1. (1)
    For any given r > 0 and x E, there exists a unique z C such that
    f ( z , y ) + 1 r y - z , J z - J x 0 , y C .
    (2.3)
     
  2. (2)
    For given r > 0 and x E, define a mapping K r f : E C by
    K r f ( x ) = z C : f ( z , y ) + 1 r y - z , J z - J x 0 , y C .
    (2.4)
     
Then, the following hold:
  1. (i)

    K r f is single-valued;

     
  2. (ii)
    K r f is a firmly nonexpansive-type mapping, i.e., for any x, y E,
    K r f ( x ) - K r f ( y ) , J K r f ( x ) - J K r f ( y ) K r f ( x ) - K r f ( y ) , J x - J y ;
     
  3. (iii)

    F K r f = E P ( f ) (the set of solutions of equilibrium problem for function f);

     
  4. (iv)

    EP(f) is closed and convex.

     
  5. (v)

    ϕ p , K r f ( x ) + ϕ K r f ( x ) , x ϕ ( p , x ) , p F K r f . .

     
For solving the generalized mixed equilibrium problem (1.1), let us assume that the following conditions are satisfied:
  1. (1)

    E is a smooth, strictly convex and reflexive Banach space and C is a nonempty closed convex subset of E

     
  2. (2)

    B: CE* is a β-inverse strongly monotone mapping;

     
  3. (3)

    F : C × C R is a bifunction satisfying the conditions (A1), (A3), (A4) in Lemma 2.5 and the following condition (A2)':

     
(A2)' for some γ ≥ 0 with γβ
F ( x , y ) + F ( y , x ) γ B x - B y 2 , x , y C ;
  1. (4)

    Ψ : C R is a lower semi-continuous and convex function.

     

Under the assumptions as above, we have the following results.

Lemma 2.6 Let E, C, B, F, Ψ satisfy the above conditions (1)-(4). Denote by
Γ ( x , y ) = F ( x , y ) + Ψ ( y ) - Ψ ( x ) + y - x , B x , x , y C .
(2.5)
For any given r > 0 and x E, define a mapping K r Γ : E C by
K r Γ ( x ) = z C : Γ ( z , y ) + 1 r y - z , J z - J x 0 , y C .
(2.6)
Then, the following hold:
  1. (i)

    K r Γ is single-valued;

     
  2. (ii)
    K r Γ is a firmly nonexpansive-type mapping, i.e., for all x,y E,
    K r Γ ( x ) - K r Γ ( y ) , J K r Γ ( x ) - J K r Γ ( y ) K r Γ ( x ) - K r Γ ( y ) , J x - J y ;
     
  3. (iii)

    F ( K r Γ ) = E P ( Γ ) = GMEP ( F , Φ , B ) ;

     
  4. (iv)

    GMEP(F, Φ, B) is closed and convex.

     
  5. (v)

    ϕ ( p , K r Γ ( x ) ) + ϕ ( K r Γ ( x ) , x ) ϕ ( p , x ) , p F ( K r Γ ) .

     

Proof. It follows from Lemma 2.5 that in order to prove the conclusions of Lemma 2.6 it is sufficient to prove that the function Γ : C × C R satisfies the conditions (A1)-(A4) in Lemma 2.5.

In fact, by the similar method as given in the proof of Lemma 2.5 in [1], we can prove that the function Γ satisfies the conditions (A1), (A3), and (A4). Now we prove that Γ also satisfies the condition (A2).

Indeed, for any x, y C, by condition (A2)' we have
Γ ( x , y ) + Γ ( y , x ) = F ( x , y ) + Ψ ( y ) - Ψ ( x ) + y - x , B x + F ( y , x ) + Ψ ( x ) - Ψ ( y ) + x - y , B y = F ( x , y ) + F ( y , x ) - x - y , B x - B y ( γ - β ) B x - B y 2 0 .

This implies that the function Γ satisfies the condition (A2). Therefore the conclusions of Lemma 2.6 can be obtained from Lemma 2.5 immediately.

In the sequel, we make use of the function V : E × E * R which is defined by
V ( x , x * ) = x 2 - 2 x , x * + x * 2 ,

for all x E and x* E*. Observe that V(x, x*) = ϕ(x, J-1 x*) for all x E and x* E*.

The following lemma is well known.

Lemma 2.7 [6] Let E be a smooth, strictly convex and reflexive Banach space with E* as its dual. Then
V ( x , x * ) + 2 J - 1 x * - x , y * V ( x , x * + y * )

for all x E and x*, y* E*.

3 Main results

In this section we shall make use of the following assumptions.
  1. (1)

    Let E be a 2-uniformly convex and uniformly smooth real Banach space with a dual E* and C be a nonempty closed and convex subset of E;

     
  2. (2)
    Let T i : CC, i = 1, 2,... be a countable family of closed and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences {ν n }, {μ n } and a strictly increasing continuous functions ς : R + R + such that ν n → 0, μ n → 0 (as n → ∞) and ζ(0) = 0. Suppose further that for each i ≥ 1, T i is a uniformly L i -Lipschitzian mapping, i.e., there exists a positive number L i > 0 such that
    T i n x - T i n y L i x - y , x , y C , n 1 ;
     
  3. (3)

    Let A i : CE*, i = 1, 2,..., N be a δ i -inverse strongly monotone mapping and denote by δ = min{δ n , i = 1,2,..., N};

     
  4. (4)

    Let B i : CE*, i = 1,2,... , M be a β i -inverse strongly monotone mappings;

     
  5. (5)

    Let F i : C × C R , i = 1 , 2 , , M be a finite family of bifunctions satisfying conditions (A1), (A3), (A4) and the following condition (A2)":

     
(A2)" For each i = 1, 2,..., M there exists γ i ≥ 0 with γ i β i such that
F i ( x , y ) + F i ( y , x ) γ i B i x - B i y 2 , x , y C ;
  1. (6)

    Let Ψ i : C R , i = 1 , 2 , , M be a finite family of lower semi-continuous convex functions.

     

We are now in a position to give the main results of this article.

Theorem 3.1 Let E , C , { T i } i = 1 , { A n } n = 1 N , { B i } i = 1 M , { F i } i = 1 M , { Ψ i } i = 1 M satisfy the above conditions (1)-(6). Suppose that
F : = i = 1 F ( T i ) n = 1 M A n - 1 ( 0 ) m = 1 GMEP ( F m , B m , Φ m )
is a nonempty and bounded subset of C. For any given x0 C, let {x n } be the sequence generated by
x 0 C 0 = C , y n = Π C J - 1 ( J x n - λ A n + 1 x n ) , z n = J - 1 α n , 0 J x n + i = 1 α n , i J T i n y n , u n = K r M , n Γ M K r M - 1 , n M - 1 K r 2 , n Γ 2 K r 1 , n Γ 1 z n , C n + 1 = { v C n : ϕ ( v , u n ) ϕ ( v , x n ) + η n } , x n + 1 = Π C n + 1 x 0 , n 0 ,
(3.1)
where
η n = ν n sup u F ς ( ϕ ( u , x n ) ) + μ n , n 1 ,
K r k , n Γ k : E C , k = 1 , 2 , , M is the mapping defined by (2.6) with Γ = Γ k , r = rk,n, and
Γ k ( x , y ) = F k ( x , y ) + y - x , B k x + Ψ k ( y ) - Ψ k ( x ) , x , y C ,
(3.2)
A n = An(modN), rk,n [d, ∞), k = 1,2,..., M, n ≥ 1 for some d > 0, 0 < λ < c 2 δ 2 , where c is the 2-uniformly convex constant of E, and for each n ≥ 1
  1. (i)

    for each n ≥ 1, i = 0 α n , i = 1 ;

     
  2. (ii)

    for each j ≥ 1, lim infn→∞αn,0αn,j> 0.

     

If F is a nonempty and bounded subset in C, then {x n } converges strongly to some point x * F .

Proof. We divide the proof of Theorem 3.1 into five steps.
  1. (I)

    Sequences {x n }, {y n } and { T i n y n } are all bounded.

     
In fact, since x n = Π C n x 0 , for any p F , from Lemma 2.2, we have
ϕ ( x n , x 0 ) = ϕ ( Π C n x 0 , x 0 ) ϕ ( p , x 0 ) - ϕ ( p , x n ) ϕ ( p , x 0 ) .

This implies that the sequence {ϕ(x n , x0)} is bounded, and so {x n } is bounded.

On the other hand, by Lemmas 2.2, 2.7, and 2.1, we have that
ϕ ( p , y n ) = ϕ ( p , Π C J - 1 ( J x n - λ A n + 1 x n ) ) ϕ ( p , J - 1 ( J x n - λ A n + 1 x n ) ) = V ( p , J x n - λ A n + 1 x n ) V ( p , ( J x n - λ A n + 1 x n ) + λ A n + 1 x n ) - 2 J - 1 ( J x n - λ A n + 1 x n ) - p , λ A n + 1 x n = V ( p , J x n ) - 2 λ J - 1 ( J x n - λ A n + 1 x n ) - p , A n + 1 x n = ϕ ( p , x n ) - 2 λ x n - p , A n + 1 x n - 2 λ J - 1 ( J x n - λ A n + 1 x n ) - x n A n + 1 x n = ϕ ( p , x n ) - 2 λ ( x n - p , A n + 1 x n - A n + 1 p ) ( since A n + 1 p = 0 ) - 2 λ J - 1 ( J x n - λ A n + 1 x n ) - x n , A n + 1 x n ϕ ( p , x n ) - 2 λ δ A n + 1 x n 2 ( by condition (3) ) + 2 λ J - 1 ( J x n - λ A n + 1 x n ) - J - 1 J x n × A n + 1 x n ϕ ( p , x n ) - 2 λ δ A n + 1 x n 2 + 4 λ 2 c 2 A n + 1 x n 2 ( by L e m m a 2 . 1 ) = ϕ ( p , x n ) + 2 λ 2 λ c 2 - δ A n + 1 x n 2
(3.3)
Thus, using the fact that λ c 2 2 δ , we have that
ϕ ( p , y n ) ϕ ( p , x n ) .
(3.4)
This shows that {y n } is also bounded. Moreover, by condition (2), { T i : C C } i = 1 is a countable family of uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences {ν n }, {μ n } and a strictly increasing continuous functions ς : R + R + such that ν n → 0, μ n → 0 (as n → ∞) and ζ(0) = 0. Therefore for each i ≥ 1 and for given p F we have
ϕ ( p , T i n y n ) ϕ ( p , y n ) + ν n ς ( ϕ ( p , y n ) ) + μ n , n 1 .
Since {y n } is bounded, this shows that, { T i n y n } is uniformly bounded. Denote by
K ̃ = sup n 0 , i 1 { x n , y n , T i n y n } < .
By the way, from the definition of {η n }, it is easy to see that
η n = ν n sup u F ς ( ϕ ( u , x n ) ) + μ n 0 ( a s n ) .
(3.5)
  1. (II)

    For each n ≥ 0, C n is a closed and convex subset of C and Ω C n .

     
It is obvious that C0 = C is closed and convex. Suppose that C n is closed and convex for some n ≥ 1. Since the inequality ϕ(v,u n ) ≤ ϕ(v, x n ) + η n is equivalent to
2 v , J x n - J u n x n 2 - u n 2 + η n ,
therefore, we have
C n + 1 = { v C n : 2 v , J x n - J u n x n 2 - u n 2 + η n } .
(3.6)

This implies that Cn+1is closed and convex. Thus for each n ≥ 0, C n is a closed and convex subset of C

Next, we prove that F C n for all n ≥ 0. Indeed, it is obvious that, F C 0 = C . Suppose F C n for some n ≥ 1. Since E is uniformly smooth, E* is uniformly convex. For any given p F C n and for any positive integers j ≥ 1 and any positive integer m = 1, 2,..., M - 1, from Lemmas 2.6(v) and 2.4, we have
ϕ ( p , u n ) = ϕ p , K r M , n Γ M K r M - 1 , n Γ M K r 2 , n Γ 2 K r 1 , n Γ 1 z n ϕ p , K r m , n m K r m - 1 , n Γ m - 1 K r 2 , n Γ 2 K r 1 , n Γ 1 z n ϕ ( p , z n ) ( by L e m m a 2 . 6 ( v ) ) = ϕ p , J - 1 α n , 0 J x n + i = 1 α n , i J T i n y n = p 2 - 2 p , α n , 0 J x n + i = 1 α n , i J T i n y n + α n , 0 J x n + i = 1 α n , i J T i n y n 2 p 2 - 2 α n , 0 p , J x n - 2 i = 1 α n , i p , J T i n y n + α n , 0 x n 2 + i = 1 α n , i T i n y n 2 - α n , 0 α n , j g J x n - J T j n y n = α n , 0 ϕ ( p , x n ) + i = 1 α n , i ϕ ( p , T i n y n ) - α n , 0 α n , j g J x n - J T j n y n α n , 0 ϕ ( p , x n ) + i = 1 α n , i { ϕ ( p , y n ) + ν n ς ( ϕ ( p , y n ) ) + μ n } - α n , 0 α n , j g J x n - J T j n y n
Combining (3.4), we have
ϕ ( p , u n ) ϕ ( p , z n ) α n , 0 ϕ ( p , x n ) + i = 1 α n , i { ϕ ( p , x n ) + ν n ς ( ϕ ( p , x n ) ) + μ n } - α n , 0 α n , j g J x n - J T i n y n ϕ ( p , x n ) + ν n sup u Ω ς ( ϕ ( u , x n ) ) + μ n - α n , 0 α n , j g J x n - J T j n y n = ϕ ( p , x n ) + η n - α n , 0 α n , j g J x n - J T j n y n ϕ ( p , x n ) + η n .
(3.8)
Hence p Cn+1and F C n for all n ≥ 0.
  1. (III)

    {x n } is a Cauchy sequence.

     
Since x n = Π C n x 0 and x n + 1 = Π C n + 1 x 0 C n + 1 C n , we have that
ϕ ( x n , x 0 ) ϕ ( x n + 1 , x 0 )

which implies that the sequence {ϕ(x n , x0)} is nondecreasing and bounded, and so lim n ϕ ( x n , x 0 )

exists. Hence for any positive integer m, using Lemma 2.2(i) we have
ϕ ( x n + m , x n ) = ϕ ( x n + m , Π C n x 0 ) ϕ ( x n + m , x 0 ) - ϕ ( x n , x 0 )
for all n ≥ 0. Since lim n ϕ ( x n , x 0 ) exists, we obtain that
ϕ ( x n + m , x n ) 0 ( n ) , m 1 .
(3.9)
Thus, by Lemma 2.3 we have that ||xn+m- x n || → 0 as n → ∞. This implies that the sequence {x n } is a Cauchy sequence in C. Since C is a nonempty closed subset of Banach space E, it is complete. Hence there exists an x* in C such that
x n x * ( n ) .
(3.10)
  1. (IV)

    We show that x * i = 1 F ( T i ) .

     
Since xn+1 Cn+1by the structure of Cn+1, we have that
ϕ ( x n + 1 , u n ) ϕ ( x n + 1 , x n ) + η n .
Again by (3.5), (3.9), and Lemma 2.3, we get that lim n x n + 1 - u n = 0 . But
x n - u n x n - x n + 1 + x n + 1 - u n .
Thus
lim n x n - u n = 0 .
(3.11)
This implies that u n x* as n → ∞. Since J is norm-to-norm uniformly continuous on bounded subsets of E we have that
lim n J x n - J u n = 0 .
(3.12)
From (3.8), (3.11), and (3.12), we have that
α n , 0 α n , j g J x n - J T j n y n ϕ ( p , x n ) - ϕ ( p , u n ) + η n x n - u n ( u n + x n ) + 2 p , J u n - J x n + η n 0 ( n ) .
(3.13)
In view of condition lim inf n α n , 0 α n , j > 0 , we have that
g J x n - J T j n y n 0 ( n ) .
(3.14)
It follows from the property of g that
J x n - J T j n y n 0 ( n ) .
(3.15)
Since x n x* and J is uniformly continuous, it yields Jx n Jx*. Hence from (3.15) we have
J T j n y n J x * ( n ) , j 1 .
(3.16)
Since E* is uniformly smooth, J-1 is uniformly continuous, it follows that
T j n y n x * ( n ) , j 1 .
(3.17)
Moreover, using inequalities (3.7) and (3.3), we obtain that
ϕ ( p , u n ) α n , 0 ϕ ( p , x n ) + i = 1 α n , i { ϕ ( p , y n ) + ν n ς ( ϕ ( p , y n ) ) + μ n } α n , 0 ϕ ( p , x n ) + i = 1 α n , i ϕ ( p , x n ) + 2 λ 2 λ c 2 - δ A n + 1 x n 2 + ν n ς ( ϕ ( p , y n ) ) + μ n ϕ ( p , x n ) + α n , j 2 λ 2 λ c 2 - δ A n + 1 x n 2 + ν n ς ( ϕ ( p , x n ) ) + μ n ϕ ( p , x n ) + α n , j 2 λ 2 λ c 2 - δ A n + 1 x n 2 + η n .
This implies that
2 α n , j λ δ - 2 λ c 2 A n + 1 x n 2 ϕ ( p , x n ) - ϕ ( p , u n ) + η n 0 ( as n ) .
By the assumption that for each j ≥ 1, lim infn→∞αn,j≥ lim infn→∞αn,0αn,j> 0, and λ ( δ - 2 λ c 2 ) > 0 , hence we have
lim n A n + 1 x n 2 = 0 .
(3.19)
This together with (3.1) shows that
lim n y n - x * = lim n Π C J - 1 ( J x n - λ A n + 1 x n ) - Π C x * lim n J - 1 ( J x n - λ A n + 1 x n ) - x * = 0
(3.20)
Furthermore, by the assumption that for each j ≥ 1, T j is uniformly L j -Lipschitz continuous, hence we have
T j n + 1 y n - T j n y n T j n + 1 y n - T j n + 1 y n + 1 + T j n + 1 y n + 1 - y n + 1 + y n + 1 - y n + y n - T j n y n ( L j + 1 ) y n + 1 - y n + T j n + 1 y n + 1 - y n + 1 + y n - T j n y n .
(3.21)
This together with (3.17) and (3.20), yields
lim n T j n + 1 y n - T j n y n = 0 .
Hence from (3.17) we have
lim n T j n + 1 y n = x * ,
i.e.,
lim n T j T j n y n = x * .
In view of (3.17) and the closeness of T j it yields that T j x* = x* for all j ≥ 1. This implies that x * j = 1 F ( T j ) .
  1. (IV)

    Now, we prove that x * n = 1 N A n - 1 ( 0 ) .

     
It follows from (3.19) that
lim n A n + 1 x n = 0 .
(3.22)
Since limn→∞x n = x*, we have that for every subsequence { x n j } j 1 { x n } n 0 , lim j x n j = x * and
lim j A n j + 1 x n j = 0 .
(3.23)
Let {n q }q ≥1 be an increasing sequence of natural numbers such that A n q + 1 = A 1 , q . Then lim p x n q - x * = 0 and
0 = lim q A n q + 1 x n q = lim q A 1 x n q .
Since A1 is γ-inverse strongly monotone, it is 1 γ -Lipschitz continuous and thus
A 1 x * = A 1 lim q x n q = lim q A 1 x n q = 0 .
Hence
x * A 1 - 1 ( 0 ) .
Continuing this process, we obtain that x * A i - 1 ( 0 ) , i = 1 , 2 , , N , i = 1, 2,..., N. Hence
x * n = 1 N A n - 1 ( 0 ) .
  1. (V)

    Next, we prove that x * m = 1 M GMEP ( F m , B m , Ψ m ) .

     
Denote
S n m = K r m , n Γ m K r m - 1 , n Γ m - 1 K r 2 , n Γ 2 K r 1 , n Γ 1 , m = 1 , 2 , , M
and S n 0 = I for all n ≥ 1. By Lemma 2.6, for any p F , we have
ϕ ( S n m z n , S n m - 1 z n ) ϕ ( p , S n m - 1 z n ) - ϕ ( p , S n m z n ) ϕ ( p , z n ) - ϕ ( p , S n m z n ) ϕ ( p , x n ) + η n - ϕ ( p , S n m z n ) ( by ( 3 . 8 ) and ( 3 . 7 ) ) ϕ ( p , x n ) + η n - ϕ ( p , u n ) ( by ( 3 . 7 ) ) 0 ( as n ) .
(3.24)
From (3.13) we have that lim n ϕ ( S n m z n , S n m - 1 z n ) = 0 . Since E is 2-uniformly convex and uniformly smooth Banach space and {z n } is bounded, we have that
lim n S n m z n - S n m - 1 z n = 0 , m = 1 , 2 , , M .
(3.25)
Next we prove that for each m = 1, 2,..., M
S n m z n x * ( as n ) .
In fact, since x n x* and u n x* (as n → ∞), if m = M then we have
lim n S n M z n - S n M - 1 z n = lim n u n - S n M - 1 z n = 0 .

This implies that S n M - 1 z n x * . By induction, the conclusion can be obtained.

Since J is norm-to-norm uniformly continuous on bounded subsets of E, from (3.25) we have
lim n J S n m z n - J S n m - 1 z n = 0 , m = 1 , 2 , , M .
(3.26)
Again since { r m , n } m = 1 M [ d , ) for some d > 0, we have that
lim n J S n m z n - J S n m - 1 z n r m , n = 0 , m = 1 , 2 , , M .
(3.27)
In the proof of Lemma 2.6 we have proved that the function Γ m , m = 1, 2,... M defined by (3.2) satisfies the condition (A1)-(A4) and
Γ m ( S n m z n , y ) + 1 r m , n y - S n m z n , J S n m z n - J S n m - 1 z n 0 , y C ,
Therefore for any y C we have
1 r m , n y - S n m z n , J S n m z n - J S n m - 1 z n - Γ m ( S n m z n , y ) Γ m ( y , S n m z n ) .
(3.28)
This implies that
Γ m ( y , S n m z n ) 1 r m , n y - S n m z n , J S n m z n - J S n m - 1 z n ( M 1 + y ) J S n m z n - J S n m - 1 z n r m , n ,
(3.29)
for some constant M1 > 0. Since the function y Γ m (x, y) is a convex and lower semi-continuous, from (3.27) and (3.29) we have
Γ m ( y , x * ) lim inf n Γ m ( y , S n m z n ) 0 , y C .
(3.30)

For any t (0, 1] and y C, then y t = ty + (1 -t)x* C. Since Γ m satisfies conditions (A1), (A4),

from (3.30) we have
0 = Γ m ( y t , y t ) t Γ m ( y t , y ) + ( 1 - t ) Γ m ( y t , x * ) t Γ m ( y t , y ) , m = 1 , 2 , M .
Deleting t and then letting t → 0, by condition (A3) we have
0 Γ m ( x * , y ) , y C , m = 1 , 2 , M ,
i.e., for each m = 1, 2,..., M we have
F m ( x * , y ) + y - x * , B m x * + Ψ m ( y ) - Ψ m ( x * ) 0 , y C .
This implies that x* GMEP(F m , B m , Ψ m ), for each m = 1, 2,..., M. Therefore, we have that
x * m = 1 M GMEP ( F m , B m , Ψ m ) .

This completes the proof of Theorem 3.1.

4 Application

It is well known that the following Hammerstian type equation
u + K f u = 0 ,
(4.1)

where K is a linear operator and f is a nonlinear Nemytskii operator, plays a crucial role in the theory of optimal control systems (see, example, [21]). Several existence and uniqueness theorems have been proved for equation (4.1) (see, for examples, [2225].

We are now ready to give an application of Theorem 3.1 to an iterative solution of the nonlinear Hammerstein type Equation (4.1).

Theorem 4.1 Let E be a real Banach space with a dual E* such that X = E x E* is a 2-uniformly convex and uniformly smooth real Banach space with norm z X 2 = u E 2 + v E * 2 , z = (u, v) X. Let C be a nonempty closed convex subset of X. Let f: EE* and K: E* → E with D(K) = f(E) = E* be continuous monotone type operators such that Equation (4.1) has a solution in E and such that the mapping A: XX* defined by
A z : = A ( u , v ) = ( f u - v , u + K v )
(4.2)
is δ-inverse strongly monotone. Let B: CX* be a β-inverse strongly monotone mappings. Let F : C × C R be a bifunction satisfying condition (A1), (A2)", (A3), (A4) in Theorem 3.1 and Ψ : C R be a lower semi-continuous and convex function. Let T: CC be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences {ν n }, {μ n } and a strictly increasing continuous functions ς : R + R + such that ν n → 0, μ n → 0 (as n → ∞) and ζ(0) = 0, and T is uniformly L- Lipschitzian. If F : = F ( T ) A - 1 ( 0 ) GMEP ( F , B , Ψ ) is a nonempty bounded subset of C, then the sequence {x n } defined by
{ x 0 C 0 = C , y n = Π C J 1 ( J x n λ A x n ) , z n = J 1 ( α n J x n + ( 1 α n J T n y n ) , u n = K r n Γ z n , C n + 1 = { v C n : ϕ ( v , u n ) ϕ ( v , x n ) + η n } , x n + 1 = Π C n + 1 x 0 , n 0 ,
(4.3)
converges strongly to some point x * F where
η n = ν n sup u F ς ( ϕ ( u , x n ) ) + μ n , n 1 ,
(4.4)
K r n Γ : E C is the mapping defined by (2.6) and
Γ ( x , y ) = F ( x , y ) + Ψ ( y ) - Ψ ( x ) + y - x , B x , x , y C ,

r n [d, ∞), n ≥ 1 for some d > 0, 0 < λ < c 2 δ 2 , where c is the 2-uniformly convex constant of E and {α n } is the sequence in (0, 1) with lim infn→∞α n (1 - α n ) > 0.

Remark 4.1 Since x * F , therefore we have x* A-1(0), i.e., Ax* = 0. Since x* = (u*,v*) for some u* E, v* E*, we have Ax* = A(u*, v*) = (fu* - v*, u* + Kv*) = (0, 0). This implies that fu* - v* = 0 and u* + Kv* = 0, i.e., u* + Kfu* = 0. Hence u* is a solution of the nonlinear Hammerstein type equation (4.1) and x n x* (as n → ∞).

Declarations

Acknowledgements

This study was supported by the Scientific Research Fund of Sichuan Provincial Education De-partment(11ZA172) and the Natural Science Foundation of Yunnan University of Finance and Economics.

Authors’ Affiliations

(1)
Department of Mathematics, Yibin University, Yibin, Sichuan, China
(2)
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, China
(3)
Department of Mathematics, Kunming University, Kunming, Yunnan, China

References

  1. Chang SS, Lee HW, Chan CK: A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach space, with applications. Nonlinear Anal 2010, 73: 2260–2270. 10.1016/j.na.2010.06.006MathSciNetView ArticleGoogle Scholar
  2. Ceng LC, Yao JC: A hybrid iterative sheme for mixed equilibrium problems and fixed point problems. J Comput Math Appl 2008, 214: 186–201. 10.1016/j.cam.2007.02.022MathSciNetView ArticleGoogle Scholar
  3. Tang JF, Chang SS: Strong convergence theorem for a generalized mixed equilibrium problem and fixed point problem for a family of infinitely nonexpansive mappings in Hilbert spaces. Pan Am Math J 2009, 19(2):75–86.MathSciNetGoogle Scholar
  4. Ofoedu EU, Malonza DM: Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type. Appl Math Comput 2011.Google Scholar
  5. Zhang SS: The generalized mixed equilibrium problem in Banach space. Appl Math Mech 2009, 30: 1105–1112. 10.1007/s10483-009-0904-6View ArticleGoogle Scholar
  6. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: AG Kartosator. Marcel Dekker, New York; 1996:15–50.Google Scholar
  7. Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J Approx Theory 2005, 134(2):257–266. 10.1016/j.jat.2005.02.007MathSciNetView ArticleGoogle Scholar
  8. Su YF, Xu HK, Zhang X: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications. Nonlinear Anal 2010, 73: 3890–3906. 10.1016/j.na.2010.08.021MathSciNetView ArticleGoogle Scholar
  9. Zhang HC, Su YF: Strong convergence of modified hybrid algorithm for quasi- ϕ -asymptotically nonexpansive mappings. Commun Korean Math Soc 2009, 24(4):539–551. 10.4134/CKMS.2009.24.4.539MathSciNetView ArticleGoogle Scholar
  10. Zegeye H, Shahzad N: Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal Theory Methods Appl 2009, 70(7):2707–2716. 10.1016/j.na.2008.03.058MathSciNetView ArticleGoogle Scholar
  11. Wattanawitoon K, Kumam P: Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi- ϕ -nonexpansive mappings. Nonlinear Anal Hybrid Syst 2009, 3(1):11–20. 10.1016/j.nahs.2008.10.002MathSciNetView ArticleGoogle Scholar
  12. Qin XL, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl Math Comput 2010, 215(11):3874–3883. 10.1016/j.amc.2009.11.031MathSciNetView ArticleGoogle Scholar
  13. Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal 2009, 70(1):45–57. 10.1016/j.na.2007.11.031MathSciNetView ArticleGoogle Scholar
  14. Chidume CE, Ofoedu EU: Approximation of common fixed point for finite families of total asymptotically nonexpansive mappings. J Math Anal Appl 2007, 333: 128–141. 10.1016/j.jmaa.2006.09.023MathSciNetView ArticleGoogle Scholar
  15. Chidume CE, Ofoedu EU: A new iteration process for approximation of common fixed point for finite families of total asymptotically nonexpansive mappings. Int J Math Math Sci 2009, 2009: 17. (Article ID 615107)MathSciNetView ArticleGoogle Scholar
  16. Alber YaI, Chidume CE, Zegeye H: Approximating fixed points of total asymptotically non-expansive mappings. Fixed Point Theory Appl 2006, 2006: 20. (Article ID 10673)MathSciNetView ArticleGoogle Scholar
  17. Xu HK: Inqualities in Banach spaces with applications. Nonlinear Anal 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleGoogle Scholar
  18. Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim 2002, 13: 938–945. 10.1137/S105262340139611XMathSciNetView ArticleGoogle Scholar
  19. Chang SS, Kim JK, Wang XR: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J Inequal Appl 2010, 2010: 14. (Article ID 869684)MathSciNetGoogle Scholar
  20. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math Stud 1994, 63: 123–145.MathSciNetGoogle Scholar
  21. Dolezale V: Monotone Operators and its Applications in Automation and Network Theory, Studies in Automation and Control 3. Elsevier, New York; 1979.Google Scholar
  22. Browder FE, De Figueiredo DG, Gupta P: Maximal monotone operators and nonlinear integral equations of Hammerstein type. Bull Am Math Soc 1970, 76: 700–705. 10.1090/S0002-9904-1970-12511-3MathSciNetView ArticleGoogle Scholar
  23. Chang SS: On Chidume's open questions and approximal solutions of nulti-valued strongly accretive mappings in Banach spaces. J Math Anal Appl 1997, 216: 94–111. 10.1006/jmaa.1997.5661MathSciNetView ArticleGoogle Scholar
  24. Chidume CE: Fixed point iterations for nonlinear Hammerstein equations involving nonex-pansive and accretive mappings. Indian J Pure Appl Math 1989, 120: 129–135.MathSciNetGoogle Scholar
  25. De Figueiredo DG, Gupta P: On the variational method for the existence of solutions to nonlinear equations of Hammerstein type. Proc Am Math Soc 1973, 40: 470–476.MathSciNetView ArticleGoogle Scholar

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© Wang et al.; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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